Bifoliated planes arise naturally in the study of Anosov flows on $3$-manifolds. To any Anosov flow on a $3$-manifold $M$, one can associate a bifoliated plane equipped with an action of the fundamental group of $M$ which encodes the topology of the flow. This thesis begins by showing left-orderability of any group acting faithfully on a bifoliated plane. We then describe the bifoliated planes associated with two families of Anosov flows which are constructed from algebraic and combinatorial data via gluing procedures. For one of these families, we show that all the resulting bifoliated planes are isomorphic. In contrast, for the other family, we show that the defining data can be recovered as a topological invariant of the bifoliated plane.